[Core] Move exception creation functionality out of ClientDiagnostics

[annelo-msft]

This needs to be addressed as a part of #23372

• Add the functionality to new type in Azure.Core
• Update generator shared source to use new Response-based APIs
• Update libraries using ClientDiagnostics to move to the new type for exception creation
• Remove functionality from ClientDiagnostics

[annelo-msft]

See nice pre-work done on this by @pakrym: https://github.com/Azure/azure-sdk-for-net/compare/main...pakrym%3Apakrym%2FException-creation-in-core?expand=1

[annelo-msft]

Investigation into services that have subclassed ResponseClassifier:

• https://github.com/Azure/azure-sdk-for-net/blob/main/sdk/identity/Azure.Identity/src/CredentialPipeline.cs
• https://github.com/Azure/azure-sdk-for-net/blob/main/sdk/identity/Azure.Identity/src/ManagedIdentitySource.cs
• https://github.com/Azure/azure-sdk-for-net/blob/main/sdk/storage/Azure.Storage.Common/src/Shared/StorageResponseClassifier.cs

[annelo-msft]

Services that have subclassed ClientDiagnostics:

• sdk/servicebus/Azure.Messaging.ServiceBus/src/Administration/ServiceBusClientDiagnostics.cs
• sdk/textanalytics/Azure.AI.TextAnalytics/src/TextAnalyticsClientDiagnostics.cs
• sdk/communication/Azure.Communication.PhoneNumbers/src/CommunicationClientDiagnostics.cs
• sdk/tables/Azure.Data.Tables/src/TablesClientDiagnostics.cs
• sdk/storage/Azure.Storage.Common/src/Shared/StorageClientDiagnostics.cs

[annelo-msft]

Work remains to move all libraries to the new implementation and remove the old functionality from ClientDiagnostics - that would be great to do in MQ.